1. 5.1 The Product Rule and Power Rules for Exponents

2. Solution: Write 2 · 2 · 2 in exponential form and evaluate.
What is the Base?
What is the Exponent?

3. Solution: Base: Exponent: Base Exponent
Evaluate. Name the base and the exponent.
Solution:
Base: Exponent:
Base Exponent
Please Note: The absence of parentheses in the first part indicate that the exponent applies only to the base 2, not −2.

4. Use the product rule for exponents.
By the definition of exponents,
Generalizing from this example
suggests the product rule for exponents.
For any positive integers m and n, a m · a n = a m + n.
(Keep the same base; add the exponents.)
Example: 62 · 65 = 67

5. Use the product rule for exponents to find each product if possible.
Solution:
The product rule does not apply.
The product rule does not apply.
Be sure you understand the difference between adding and multiplying exponential expressions. For example,
but

6. Use the rule (am)n = amn.
We can simplify an expression such as (83)2 with the product rule for exponents.
The exponents in (83)2 are multiplied to give the exponent in 86.
This example suggests the first power rule for exponents.
For any positive number integers m and n, (am)n = amn.
(Raise a power to a power by multiplying exponents.)
Example:

7. Simplify.
Solution:
Be careful not to confuse the product rule, where 42 · 43 = 42+3 = 45 =1024
with the power rule (a) where (42)3 = 42 · 3 = 46 = 4096.

8. Use the rule (ab)m = ambm.
We can rewrite the expression (4x)3 as follows.
This example suggests the second power rule for exponents.
For any positive integer m, (ab)m = ambm.
(Raise a product to a power by raising each factor to the power.)
Example:

9. Simplify.
Solution:
Use the second power rule only if there is one term inside parentheses.
Power rule does not apply to a sum. For example,
, but

10. 2^15
X^10
9x^2y^2
48m^8p^12

11. Use the rule
Since the quotient can be written as we use this fact and second power rule to get the third power rule for exponents.
For any positive integer m,
Example:

12. Simplify.
Solution:
In general, 1n = 1, for any integer n.

13. Rules of Exponents

14. Simplify
Solution:
4x^2/625
25k^6/9
-27x^11y^10

15. Find an expression that represents the area of the figure.
Solution:

16. Homework
5.1: 1 – 87 EOO